Improved accuracy for the approximate factorization of parabolic equations

A general procedure to construct alternating direction implicit (ADI) schemes for multidimensional problems was originated by Beam and Warming, using the method of approximate factorization. The technique which can be combined with a high-order linear multistep (LM) method introduces a factorization error that is of order two in the time step Δt. Thus, the approximate factorization method imposes a second-order temporal accuracy limitation independent of the accuracy of the LM method chosen as the time differencing approximation. We introduce a correction term to the right-hand side of a factored scheme to increase the order of the factorization error in Δt, and recover the temporal order of the original scheme. The method leads in particular to the modified ADI scheme proposed by Douglas and Kim. A convergence proof is given for the improved scheme based on the BDF2 method.